OFFSET
0,3
COMMENTS
Equals INVERT transform of (1, 2, 0, 1, 0, 1, 0, 1, ...). - Gary W. Adamson, Apr 28 2009
The sequence is also the INVERT transform of the aerated odd-indexed Fibonacci numbers (i.e., of (1, 0, 2, 0, 5, 0, ...)). Sequence A124400 is the INVERT transform of the aerated even-indexed Fibonacci numbers. - Gary W. Adamson, Feb 07 2014
a(n) is the number of tilings of a 4 X 2n rectangle into L tetrominoes (no reflections, only rotations). - Nicolas Bělohoubek, Jan 21 2025
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
Nicolas Bělohoubek and Antonín Slavík, L-Tetromino Tilings and Two-Color Integer Compositions, Univ. Karlova (Czechia, 2025). See p. 3.
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).
FORMULA
G.f.: (1-x^2)/(1 - x - 3x^2 + x^3 + x^4). - Philippe Deléham, Jan 21 2012
a(n) = a(n-1) + 3*a(n-2) - a(n-3) - a(n-4), a(0)=1, a(1)=1, a(2)=3, a(3)=5. - Philippe Deléham, Jan 21 2012
a(n) = Sum_{m=0..ceiling(n/2)} binomial(n-m,n-2*m)*Fibonacci(n-2*m+1). - Vladimir Kruchinin, Jan 26 2013
From Nicolas Bělohoubek, Jan 21 2025: (Start)
a(n) = Sum_{m=1..4} (alpha_m * x_m^n). For x_m and alpha_m values see "L-tetromino tilings" article in links.
a(2*n) = A166482(n). (End)
EXAMPLE
a(4) = 12 = 5 + 0 + 6 + 0 + 1.
MATHEMATICA
LinearRecurrence[{1, 3, -1, -1}, {1, 1, 3, 5}, 50] (* Paolo Xausa, Jan 28 2025 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jun 28 2007
EXTENSIONS
a(10)-a(30) from Philippe Deléham, Jan 21 2012
STATUS
approved